Finite Group Databasebeta

← C109×#12770 #42130764

Subgroups — C109×#12770 #42130764

Order 27904.

752 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3551C211yesmaximal, characteristic
2C109×#2132C211yesmaximal, characteristic, complemented
3C109×#2123C211yesmaximal, characteristic, complemented
4C109×#2122C211yesmaximal, characteristic, complemented
5C109×#1915C211yesmaximal, characteristic, complemented
6C109×#1914C211yesmaximal, characteristic, complemented
7C109×#1884C211yesmaximal, characteristic, complemented
8#568×C109C2211yescharacteristic
9#563×C109—21no—
10#560×C109C412yescomplemented
11#555×C109—21no—
12#553×C109—21no—
13#552×C109—21no—
14#551×C109C412yescomplemented
15#548×C109—21no—
16#546×C109—21no—
17#540×C109—22no—
18#538×C109—22no—
19#537×C109—21no—
20#536×C109C2211yescharacteristic
21#535×C109C2211yescharacteristic
22#532×C109C2211yescharacteristic
23#529×C109—21no—
24#421×C109—21no—
25#413×C109C2211yescharacteristic
26#413×C109C2211yescharacteristic
27C2×#101×C109—21no—
28C2×#100×C109—21no—
29#410×C109C2211yescharacteristic
30#356×C109—21no—
31#355×C109—21no—
32#352×C109—22no—
33AΣL(1,𝔽16)×C109—22no—
34#344×C109—21no—
35#344×C109—21no—
36#339×C109—21no—
37C2×#41×C109—21no—
38C2×#41×C109D411yescharacteristic
39C2×#41×C109C2311yescharacteristic
40C22×Q8×C109D411yescharacteristic
41C22×D4×C109D411yescharacteristic
42#131×C109—42no—
43C2×M16×C109—21no—
44C2×M16×C109—21no—
45C2×M16×C109—21no—
46C2×M16×C109—21no—
47C2×M16×C109—21no—
48C4⋊Q8×C109—21no—
49C4⋊Q8×C109D411yescharacteristic
50#127×C109—21no—
51#127×C109C2×C411yescharacteristic
52#126×C109—22no—
53#126×C109—22no—
54#126×C109—22no—
55#125×C109—21no—
56#125×C109—22no—
57#124×C109—21no—
58#124×C109—22no—
59#124×C109—21no—
60#124×C109—21no—
61#124×C109C2×C411yescharacteristic
62#124×C109D411yescharacteristic
63#123×C109—22no—
64#123×C109—22no—
65#123×C109—22no—
66#123×C109—22no—
67#123×C109C2×C412yes—
68#123×C109D412yes—
69C22⋊Q8×C109—21no—
70C22⋊Q8×C109—22no—
71C22⋊Q8×C109—22no—
72C22⋊Q8×C109—21no—
73C22⋊Q8×C109—21no—
74#121×C109—22no—
75#121×C109—22no—
76#121×C109—21no—
77#121×C109—22no—
78#121×C109—22no—
79#121×C109—21no—
80#121×C109—21no—
81#121×C109—22no—
82#121×C109—21no—
83#121×C109—21no—
84#121×C109—22no—
85#120×C109—22no—
86#120×C109C2×C412yes—
87#120×C109D412yes—
88C4×Q8×C109—21no—
89C4×D4×C109—22no—
90C4×D4×C109—21no—
91C4×D4×C109—22no—
92C4×D4×C109—22no—
93C4×D4×C109—21no—
94C4×D4×C109—21no—
95#117×C109—22no—
96#117×C109—22no—
97#117×C109—21no—
98#117×C109D411yescharacteristic
99#117×C109D411yescharacteristic
100C2×C4⋊C4×C109—21no—
101C2×C22⋊C4×C109—22no—
102C2×C42×C109D411yescharacteristic
103#104×C109—42no—
104#101×C109—21no—
105#101×C109—21no—
106#101×C109—42no—
107#100×C109—21no—
108#100×C109—21no—
109#100×C109—42no—
110#99×C109—21no—
111#99×C109—21no—
112C24×C109C22⋊C412yes—
113#41×C109—41no—
114#41×C109—41no—
115#41×C109—41no—
116#41×C109—41no—
117#41×C109—81no—
118#41×C109—81no—
119C2×Q8×C109—21no—
120C2×Q8×C109—21no—
121C2×Q8×C109—41no—
122C2×Q8×C109—41no—
123C2×Q8×C109C22⋊C411yescharacteristic
124C2×Q8×C109C2×D411yescharacteristic
125C2×D4×C109—22no—
126C2×D4×C109—22no—
127C2×D4×C109—21no—
128C2×D4×C109—22no—
129C2×D4×C109—21no—
130C2×D4×C109—22no—
131C2×D4×C109—21no—
132C2×D4×C109—21no—
133C2×D4×C109—21no—
134C2×D4×C109—41no—
135C2×D4×C109—41no—
136C2×D4×C109C22⋊C411yescharacteristic
137C2×D4×C109C22×C411yescharacteristic
138C2×D4×C109C2×D411yescharacteristic
139C2×D4×C109C2×D411yescharacteristic
140C22×C4×C109—22no—
141C22×C4×C109—22no—
142C22×C4×C109—22no—
143C22×C4×C109—21no—
144C22×C4×C109—21no—
145C22×C4×C109C22⋊C412yes—
146C22×C4×C109C2×D411yescharacteristic
147C22×C4×C109C2×D411yescharacteristic
148C22×C4×C109C2×D411yescharacteristic
149M16×C109—22no—
150M16×C109—22no—
151M16×C109—22no—
152M16×C109—41no—
153M16×C109—41no—
154M16×C109—41no—
155M16×C109—41no—
156M16×C109—41no—
157C2×C8×C109—41no—
158C2×C8×C109—41no—
159C2×C8×C109—41no—
160C2×C8×C109—41no—
161C2×C8×C109—41no—
162C4⋊C4×C109—22no—
163C4⋊C4×C109—22no—
164C4⋊C4×C109—22no—
165C4⋊C4×C109—22no—
166C4⋊C4×C109—22no—
167C4⋊C4×C109—22no—
168C4⋊C4×C109—22no—
169C4⋊C4×C109—21no—
170C4⋊C4×C109—21no—
171C4⋊C4×C109—21no—
172C4⋊C4×C109—21no—
173C4⋊C4×C109—41no—
174C22⋊C4×C109—22no—
175C22⋊C4×C109—22no—
176C22⋊C4×C109—22no—
177C22⋊C4×C109—22no—
178C22⋊C4×C109—22no—
179C22⋊C4×C109—21no—
180C22⋊C4×C109—22no—
181C22⋊C4×C109—21no—
182C22⋊C4×C109—22no—
183C22⋊C4×C109—22no—
184C22⋊C4×C109—21no—
185C22⋊C4×C109—21no—
186C22⋊C4×C109—22no—
187C22⋊C4×C109—41no—
188C22⋊C4×C109C22⋊C412yes—
189C22⋊C4×C109C22⋊C412yes—
190C42×C109—22no—
191C42×C109—21no—
192C42×C109—21no—
193C42×C109—21no—
194C42×C109—41no—
195C42×C109C22⋊C411yescharacteristic
196C42×C109C22⋊C411yescharacteristic
197C23×C109—22no—
198C23×C109—21no—
199C23×C109—42no—
200C23×C109—82no—
201C23×C109C2×C22⋊C411yescharacteristic
202C23×C109C2×C22⋊C411yescharacteristic
203C23×C109#12011yescharacteristic
204Q8×C109—41no—
205Q8×C109—41no—
206Q8×C109—41no—
207Q8×C109—41no—
208Q8×C109—81no—
209D4×C109—41no—
210D4×C109—41no—
211D4×C109—41no—
212D4×C109—41no—
213D4×C109—41no—
214D4×C109—41no—
215D4×C109—81no—
216D4×C109—81no—
217D4×C109—82no—
218D4×C109—82no—
219D4×C109—82no—
220D4×C109—81no—
221D4×C109—81no—
222D4×C109—82no—
223C2×C4×C109—21no—
224C2×C4×C109—21no—
225C2×C4×C109—21no—
226C2×C4×C109—21no—
227C2×C4×C109—21no—
228C2×C4×C109—21no—
229C2×C4×C109—21no—
230C2×C4×C109—22no—
231C2×C4×C109—21no—
232C2×C4×C109—22no—
233C2×C4×C109—22no—
234C2×C4×C109—21no—
235C2×C4×C109—41no—
236C2×C4×C109—41no—
237C2×C4×C109—42no—
238C2×C4×C109—82no—
239C2×C4×C109—81no—
240C2×C4×C109—82no—
241C2×C4×C109—82no—
242C2×C4×C109—81no—
243C2×C4×C109C2×C22⋊C411yescharacteristic
244C2×C4×C109#12011yescharacteristic
245C2×C4×C109#12011yescharacteristic
246C2×C4×C109#12011yescharacteristic
247C8×C109—42no—
248C8×C109—81no—
249C8×C109—81no—
250C22×C109—21no—
251C22×C109—21no—
252C22×C109—21no—
253C22×C109—41no—
254C22×C109—41no—
255C22×C109—41no—
256C22×C109—41no—
257C22×C109—42no—
258C22×C109—41no—
259C22×C109—42no—
260C22×C109—41no—
261C22×C109—42no—
262C22×C109—162no—
263C22×C109#37911yescharacteristic
264C4×C109—21no—
265C4×C109—21no—
266C4×C109—21no—
267C4×C109—41no—
268C4×C109—42no—
269C4×C109—42no—
270C4×C109—41no—
271C4×C109—41no—
272C4×C109—42no—
273C4×C109—161no—
274C4×C109B(4,𝔽2)11yescharacteristic
275C4×C109#45811yescharacteristic
276#12770C10911yesmaximal, characteristic, direct product factor, Sylow
277C2×C109—21no—
278C2×C109—41no—
279C2×C109—41no—
280C2×C109—41no—
281C2×C109—81no—
282C2×C109—82no—
283C2×C109#179011yescharacteristic
284#3551C2×C10911yescharacteristic
285#2132C2×C10911yescharacteristic, complemented
286#2123C2×C10911yescharacteristic, complemented
287#2122C2×C10911yescharacteristic, complemented
288#1915C2×C10911yescharacteristic, complemented
289#1914C2×C10911yescharacteristic, complemented
290#1884C2×C10911yescharacteristic, complemented
291C109#1277011yescharacteristic, direct product factor, Sylow
292#568C22×C10911yescharacteristic
293#563—21no—
294#560C4×C10912yescomplemented
295#555—21no—
296#553—21no—
297#552—21no—
298#551C4×C10912yescomplemented
299#548—21no—
300#546—21no—
301#540—22no—
302#538—22no—
303#537—21no—
304#536C22×C10911yescharacteristic
305#535C22×C10911yescharacteristic
306#532C22×C10911yescharacteristic
307#529—21no—
308#421—21no—
309#413C22×C10911yescharacteristic
310#413C22×C10911yescharacteristic
311C2×#101—21no—
312C2×#100—21no—
313#410C22×C10911yescharacteristic
314#356—21no—
315#355—21no—
316#352—22no—
317AΣL(1,𝔽16)—22no—
318#344—21no—
319#344—21no—
320#339—21no—
321C2×#41—21no—
322C2×#41D4×C10911yescharacteristic
323C2×#41C23×C10911yescharacteristic
324C22×Q8D4×C10911yescharacteristic
325C22×D4D4×C10911yescharacteristic
326#131—42no—
327C2×M16—21no—
328C2×M16—21no—
329C2×M16—21no—
330C2×M16—21no—
331C2×M16—21no—
332C4⋊Q8—21no—
333C4⋊Q8D4×C10911yescharacteristic
334#127—21no—
335#127C2×C4×C10911yescharacteristic
336#126—22no—
337#126—22no—
338#126—22no—
339#125—21no—
340#125—22no—
341#124—21no—
342#124—22no—
343#124—21no—
344#124—21no—
345#124C2×C4×C10911yescharacteristic
346#124D4×C10911yescharacteristic
347#123—22no—
348#123—22no—
349#123—22no—
350#123—22no—
351#123C2×C4×C10912yes—
352#123D4×C10912yes—
353C22⋊Q8—21no—
354C22⋊Q8—22no—
355C22⋊Q8—22no—
356C22⋊Q8—21no—
357C22⋊Q8—21no—
358#121—22no—
359#121—22no—
360#121—21no—
361#121—22no—
362#121—22no—
363#121—21no—
364#121—21no—
365#121—22no—
366#121—21no—
367#121—21no—
368#121—22no—
369#120—22no—
370#120C2×C4×C10912yes—
371#120D4×C10912yes—
372C4×Q8—21no—
373C4×D4—22no—
374C4×D4—21no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—21no—
378C4×D4—21no—
379#117—22no—
380#117—22no—
381#117—21no—
382#117D4×C10911yescharacteristic
383#117D4×C10911yescharacteristic
384C2×C4⋊C4—21no—
385C2×C22⋊C4—22no—
386C2×C42D4×C10911yescharacteristic
387#104—42no—
388#101—21no—
389#101—21no—
390#101—42no—
391#100—21no—
392#100—21no—
393#100—42no—
394#99—21no—
395#99—21no—
396C24C22⋊C4×C10912yes—
397#41—41no—
398#41—41no—
399#41—41no—
400#41—41no—
401#41—81no—
402#41—81no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—41no—
406C2×Q8—41no—
407C2×Q8C22⋊C4×C10911yescharacteristic
408C2×Q8C2×D4×C10911yescharacteristic
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4C22⋊C4×C10911yescharacteristic
421C2×D4C22×C4×C10911yescharacteristic
422C2×D4C2×D4×C10911yescharacteristic
423C2×D4C2×D4×C10911yescharacteristic
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4C22⋊C4×C10912yes—
430C22×C4C2×D4×C10911yescharacteristic
431C22×C4C2×D4×C10911yescharacteristic
432C22×C4C2×D4×C10911yescharacteristic
433M16—22no—
434M16—22no—
435M16—22no—
436M16—41no—
437M16—41no—
438M16—41no—
439M16—41no—
440M16—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—41no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—21no—
464C22⋊C4—22no—
465C22⋊C4—21no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—22no—
471C22⋊C4—41no—
472C22⋊C4C22⋊C4×C10912yes—
473C22⋊C4C22⋊C4×C10912yes—
474C42—22no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—41no—
479C42C22⋊C4×C10911yescharacteristic
480C42C22⋊C4×C10911yescharacteristic
481C23—22no—
482C23—21no—
483C23—42no—
484C23—82no—
485C23C2×C22⋊C4×C10911yescharacteristic
486C23C2×C22⋊C4×C10911yescharacteristic
487C23#120×C10911yescharacteristic
488Q8—41no—
489Q8—41no—
490Q8—41no—
491Q8—41no—
492Q8—81no—
493D4—41no—
494D4—41no—
495D4—41no—
496D4—41no—
497D4—41no—
498D4—41no—
499D4—81no—
500D4—81no—
501D4—82no—
502D4—82no—
503D4—82no—
504D4—82no—
505D4—81no—
506D4—81no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4C2×C22⋊C4×C10911yescharacteristic
528C2×C4#120×C10911yescharacteristic
529C2×C4#120×C10911yescharacteristic
530C2×C4#120×C10911yescharacteristic
531C8—42no—
532C8—81no—
533C8—81no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—42no—
542C22—41no—
543C22—42no—
544C22—41no—
545C22—42no—
546C22—162no—
547C22#379×C10911yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—41no—
552C4—42no—
553C4—42no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4—161no—
558C4B(4,𝔽2)×C10911yescharacteristic
559C4#458×C10911yescharacteristic
560C2—21no—
561C2—41no—
562C2—41no—
563C2—41no—
564C2—82no—
565C2—81no—
566C2C109×#179011yescharacteristic
subgroupG/H
center, socleC2×C109#1790
commutatorC2×D4C22×C4×C109
Frattini, cosocleC2×#41C23×C109
prime radicalC109×#3551C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C2×C4×C109 ◁ C2×#41×C109 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C109×#2132C2semidirect
3. C109×#2123C2semidirect
4. C109×#2122C2semidirect
5. C109×#1915C2semidirect
6. C109×#1914C2semidirect
7. C109×#1884C2semidirect
10. #560×C109C4semidirect
14. #551×C109C4semidirect
276. #12770C109direct (N × H)
285. #2132C2×C109semidirect
286. #2123C2×C109semidirect
287. #2122C2×C109semidirect
288. #1915C2×C109semidirect
289. #1914C2×C109semidirect
290. #1884C2×C109semidirect
291. C109#12770direct (N × H)
294. #560C4×C109semidirect
298. #551C4×C109semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C109×#35512
C109×#21322
C109×#21232
C109×#21222
C109×#19152
C109×#19142
C109×#18842
#12770109